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Analysis of Planetary Equations!
By!
Ian Beardsley!
Copyright © 2021 by Ian Beardsley"
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"
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Table of Contents!
Introduction……………………….4!
Analysis……………………………11!
The Protoplanetary Disc…………14!
Weird Arithmetic…………………..17!
The AI Motif………………………..22!
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Introduction
The Titius-Bode rule describes the distribution of the planets around the sun, but starts to fail
pretty bad at Neptune. I have devised several schemes in terms of various factors to describe
the distribution of the planets. One prominent feature in these instances is that two planets
stick out in their characteristics, Venus and Mars. Other than Mercury and Earth, these are the
solid, terrestrial planets; the rest are gas giants. They are on either side of the Earth and the
closest to it. Venus is closer to the Sun than our Earth and Mars is further. Venus and Mars
have always been of great interest to us. Venus to the Russians as they have sent several
probes to it and Mars to the United States as we have sent several roaming landers to it. This
is interesting, Venus is often called the sister planet to Earth, because it is of similar size and
mass as Earth. It would seem she was once much cooler but underwent a runaway
greenhouse effect making it too hot to be habitable, hence its interest to the Russians, and
Mars while further from the sun and colder than the Earth, it is still habitable, and thus
colonizable, and hence its interest to the United States. Interestingly in our search for life on
Mars, we recently found hints of microbial life in the atmosphere of Venus (2020). Often I don’t
include Mercury because it is so small and not very massive so it didn’t contribute much to the
nature of the protoplanetary disc from which the planets formed, and mostly because its orbit
is so eccentric that it might not make sense to consider its average orbital distance. The
purpose of this paper is to explore the equations of Venus and Mars, as they are pivotal in the
importance of human destine. I also pull perfect expressions out of different schemes to begin
to make a perfect table. The schemes have theoretical implications in mathematics and
physics, but this is treated in other works. It would seem running through the thread of solar
system origin and structure is the recurring motif of artificial intelligence logic circuitry theory.
Guessing a solution to a weird problem requires weird mathematics. I create weird arithmetic. It
would seem the distribution of the planets have an AI motif. At this point I suggest that
it defies a single algebraic expression because it may be that it uses logic gate
arithmetic, which does not follow arithmetic as is done in mathematics, but rather in
creating logic circuits with the seven basic gates."
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Characterizing the distribution of the planets around the sun seems to defy a mathematical
expression. Even the Titius-Bode rule falls apart pretty badly at Neptune.!
The Titius-Bode Rule is:!
!
!
Which produces the orbits of the planets in astronomical units as such in AU:!
r = 0.4 + (0.3)2
n
n = − ∞,0,1,2,…
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However, I find if we break-up the solar system into two parts; planets interior to the asteroid
belt, and planets exterior to the asteroid belt, quite an interesting pattern forms:"
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I have devised a scheme for the planets in terms of the golden ratio conjugate phi and
Euler’s number e:!
!
!
!
!
!
!
!
So that for the planets exterior to the asteroid belt or where
is is the asteroid belt ( ). .
Which is the solution to the differential equation
Where have we seen this? In computer science.
means
Where n is the number of bits in a number N in binary. We write in binary
0=0
1=1
10=2
11=3
100=4
101=5
110=6
111=7
(ϕ)
(1 − ϕ)e
ϕ
= 0.7AU = Venus
ϕ
2
e
(2−ϕ)
= 1.52 = M ars
2ϕe
(2−ϕ)
= 4.9 = Jupiter
4ϕe
(2−ϕ)
= 10 = Sat ur n
8ϕe
(2−ϕ)
= 19.69 = Uranus
16ϕe
(2−ϕ)
= 39.38 = Nept une
P
n
= 2
n
ϕe
(2−ϕ)
P
n
= c2
n
c = ϕe
(2−ϕ)
= 2.461
c2
0
= c = 2.461
P
0
ϕ =
5 − 1
2
P
1
= Jupiter
P
2
= Sat ur n
P
3
= Ura nu s
P
4
= Nept u ne
d
2
y
dn
2
− 2log(2)
dy
dn
+ log
2
(2)y = 0
log
2
N = n
2
n
= N
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1000=8
1001=9
1010=10
1011=11
1100=12
1101=13
1110=14
1111=15
10000=16…
But what is interesting about this?
You can’t have a fractional number of bits, thus the spectrum is quantized according to whole
number solutions of
But so are the planets given by
!
!
!
!
Meaning, since we have 2, 4, 8. 16 that the planets are quantized into whole number orbits
according to computer binary with Jupiter as 2, Saturn as 4, Uranus as 8, and Neptune as 16 if
we do it in terms of Euler’s number, e and the golden ratio conjugate, .!
That is, 2=10, 4=100, 8=1000, 16=10000!
Are all zeros after a one."
log
2
3 = n
n =
log3
log2
= 1.5847
2
n
= N
P
n
= c2
n
2ϕe
(2−ϕ)
= 4.9 = Jupiter
4ϕe
(2−ϕ)
= 10 = Sat ur n
8ϕe
(2−ϕ)
= 19.69 = Uranus
16ϕe
(2−ϕ)
= 39.38 = Nept une
ϕ
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However, if we look at the artificial intelligence (semiconductor elements) silicon Si and
germanium Ge by molar mass, the planets, we have the the 3/2 appears in the ratio of the
terrestrial planets Earth, and Mars, which is 1.52 which I said was in earlier work was:!
!
However, we used values of Si=28.09 and Ge=72.61. Recent measurements of Ge give a
slightly modified value for Ge. It is 72.64. Let us compute these two equations since they are,
as we are guessing the most important, in terms of Si, to two places after the decimal, and for
Ge using the most recent value:!
Ge=72.64 g/mol!
Si=28.0855 g/mol!
Thus,…!
!
We see it is still accurate to two places after the decimal. Let us now look at Mars,…!
!
It needs to be close to 1.52AU and we see we made a mistake with the equation to begin with.
We find the answer is in averaging this with the difference of squares in the denominator with
the square of the difference in denominator, that is!
with !
Which gives our modified Mars equation:!
!
We compute its accuracy:!
!
This has an accuracy of:!
!
1.52 =
2SiGe
Ge
2
− Si
2
1
72.64
2
2(28.0855)(72.64) +
28.0855
3
72.64
1 +
28.0855
2
72.64
2
= 0.722995806
2SiGe
Ge
2
− Si
2
=
2(28.0855)(72.64)
72.64
2
− 28.0855
2
= 0.909
2SiGe
Ge
2
− Si
2
2SiGe
(Ge − Si)
2
m ars =
2SiGe
2
(Si − Ge)
2
(Si + Ge)
m ars =
2(28.0855)(72.64)
2
199950.5396
= 1.48AU
1.48/1.52 = 0.973684
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97%!
The table for the planets in terms of AI elements is:!
"
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Analysis
In all the versions of the equations for the distributions of the planets that I have devised in
terms of different factors, it is always Venus and Mars that stand out. We begin with AI:!
Earth is:!
!
We invert that to obtain Jupiter:!
!
The rest follow from it:!
Saturn: !
Uranus: !
Neptune: !
Since Mercury is the first planet it is appropriately the simplest mathematical expression for AI
semiconductor elements Si and Ge, being Mercury: Si/Ge.!
Now to Venus and Mars:!
Venus: !
Mars: !
The Venus equation here is very accurate giving 0.72 AU to two places after the decimal, and I
see it as very interesting, if not elegant in its form.!
In the following scheme:!
!
SiGe
(Ge
2
− 2SiGe + Si
2
)
Si
2
+ 2SiGe + Ge
2
SiGe
2(Si
2
+ 2SiGe + Ge
2
)
SiGe
4(Si
2
+ 2SiGe + Ge
2
)
SiGe
6(Si
2
+ 2SiGe + Ge
2
)
SiGe
1
Ge
2
2SiGe +
Si
3
Ge
1 +
Si
2
Ge
2
2SiGe
2
(Si − Ge)
2
(Si + Ge)
1
4
n +
1
4
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With Venus=2 is pretty accurate (0.75) actual (0.72).!
But the Earth equation is perfect:!
!
Earth is n=3 yields 1.00 AU. Now the equation breaks the pattern at Mars (n=4):!
!
Produces 1.5 very close to the actual value of 1.52. In my scheme where the orbits are
quantized in terms of whole numbers using and e, the golden ratio conjugate and Euler’s
number (whole numbers 2, 4. 8, 16,…) Mars is perfect as:!
!
The whole number quantization is beginning after this at Jupiter , Saturn ,
Uranus ,…!
In my Taylor expansion of!
!
!
Mercury is 0.5, Venus is 0.7, Earth is 1.00 (perfect), making Mars approximately as 1.412."
1
4
n +
1
4
1
4
n +
1
2
ϕ
ϕ
2
e
(2−ϕ)
= 1.52
2
n
2ϕe
(2−ϕ)
4ϕe
(2−ϕ)
8ϕe
(2−ϕ)
2n − 3
2n − 6
=
(
1
2
,
n
6
,
n
2
18
,
n
3
54
,
n
4
162
)
n = 18
2
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Making A Perfect Table!
"
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The Protoplanetary Disc
Indeed why look at AI elements to describe the solar system? (Other than the planets can be
whole number quantized in terms of digital binary.)!
"
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!
Taking the protoplanetary disc as a thin disc we integrate from its center to the edge, with
density decreasing linearly to zero at the edge. Thus, if the density function is given by!
!
And, our integral is!
!
!
!
The mass of the solar system adding up all the planets yields!
!
That accounts for!
82% of the mass of the solar system not including the sun, that is, of the
protoplanetary disc surrounding the sun.!
Using germanium alone, we get,!
!
If we weight the mixture of silicon and germanium as 1/3 and 2/3, then we have!
!
Which is very close.!
93%!
This is all very good, because I only used the planets and asteroids.!
Si + Ge
2
=
2.33 + 5.323
2
= 3.8265g/cm
3
ρ(r) = ρ
0
(
1 −
r
R
)
M =
∫
2π
0
∫
R
0
ρ
0
(
1 −
r
R
)
rdrdθ
M =
πρ
0
R
2
3
π(3.8265)(7.4 × 10
14
)
2
3
= 2.194 × 10
30
gra m s
M = 2.668 × 10
30
gra m s
2.194
2.668
100 =
π(5.323)(7.4 × 10
14
)
2
3
= 3.05 × 10
30
gra m s
π(4.32467)(7.4 × 10
14
)
2
3
= 2.48 × 10
30
gra m s
2.48
2.668
100 =
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Weighting silicon and germanium as 1/4 and 3/4 we have!
!
Which accounts for!
98%!
Of the mass of the solar system (very accurate).!
This mixture of 1/4 to 3/4 is a combination that exists in the Earth atmosphere which is
approximately the mixture of oxygen to nitrogen. The earth atmosphere can be considered a
mixture of chiefly O2 and N2 in these proportions:!
Air is about 25% oxygen gas (O2) by volume and 75% nitrogen gas (N2) by volume meaning
the molar mass of air as a mixture is:!
!
By molar mass the ratio of air to H20 (water) is about the golden ratio:!
!
I am not saying the solar system was a thin disk with density of the weighted mean somewhere
between silicon and germanium, but that it can be modeled as such, though if the
protoplanetary disk that eclipses epsilon aurigae every 27 years is any indication of what a
protoplanetary cloud is like, it is a thin disk in the sense that it is about 1 AU thick and 10 AU in
diameter. This around a star orbiting another star.!
π(4.4.57475)(7.4 × 10
14
)
2
3
= 2.623 × 10
30
gra m s
2.623
2.668
100 =
0.25O
2
+ 0.75N
2
≈ air
air
H
2
O
≈ Φ
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Weird Arithmetic!
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Since the planets seemed to defy a single expression for their distribution, I decided something
weird was going on. So, in order to find an algorithm for them, I decided to create a weird
arithmetic. To do this I decided to change the order of operations in an expression, and chose
the equation of a straight line:!
y=mx+b!
I decided that the slope m should be 2 because for!
!
We use the identity!
!
Because!
!
= !
I decided that b should be -3 because the Earth is the third planet. So to do weird arithmetic I
thought in!
y=2x-3!
That, the weird evaluation would be done by doing addition first so we take x-3 first then do
multiplication yielding!
!
Which is!
!
I then compared the regular arithmetic to the weird arithmetic by evaluating!
!
In a Taylor expansion yielding:!
!
∫
x
2
d x
1
2
+ 2
2
+ 3
2
+ … + n
2
=
n(n + 1)(2n + 1)
6
∫
4
1
x
2
d x = lim
n→∞
n
∑
i=1
(
4i
n
)
2
4
n
lim
n→∞
64
n
3
(
n(n + 1)(2n + 1)
6
)
=
64
3
2n − 3 → 2(n − 3)
2(n − 3) = 2n − 6
2n − 3
2n − 6
2n − 3
2n − 6
=
1
2
−
n
6
−
n
2
18
−
n
3
54
−
n
4
162
+ O(n
5
)
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Since 1/2=0.5 is approximately Mercury orbit (0.4) I took the third term as Earth which since
it needed to be 1.00AU I chose n as which gave the orbits of the planets:!
!
!
!
!
!
!
Then after the asteroids it skips to n to the sixth for Jupiter:!
!
!
!
!
Thus the equation for the planets is:!
!
Which is!
!
n
2
18
n = 18
n = 18 = 4.24264
1
2
= 0.5 = mercur y = 0.4AU
n
6
= 0.7 = venus = 0.72AU
n
2
18
= 1.00 = ear th = 1.00AU
n
3
54
= 1.412 ≈ 2 = m ars = 1.52AU
n
4
162
= 1.999 ≈ 2 = asteroid s = 2AU − 3AU
n
6
1458
= 5.1 = jupiter = 5.2AU
n
9
39366
= 11.31AU = sat ur n = 9.5AU
n
11
354294
= 22.624AU = uranus = 19AU
n
12
1062882
= 32AU = neptune = 30AU
(
1
2
,
n
6
,
n
2
18
,
n
3
54
,
n
4
162
)
P
i
=
(
1
2
,
n
i
2 ⋅ 3
i
, . . .
)
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i=(1, 2, 3, 4, 5..)!
!
!
.!
.!
.!
Starting with Venus as . The Taylor expansion is:!
Comparing regular arithmetic to weird arithmetic:
P
1
=
n
1
2 ⋅ 3
1
=
n
6
P
2
=
n
2
2 ⋅ 3
2
=
n
2
18
P
1
∞
∑
n= 0
f
n
(a)
n!
(x − a)
n
= f (a) + f′ (a)(x − a) +
f′ ′ (a)
2!
(x − a)
2
+ …
2x − 3 → 2(x − 3)
2x − 3
2x − 6
d
dx
(2x − 3)
(2x − 6)
=
f′ (x)g(x) − g′ (x)f (x)
g(x)
2
f′ (x) =
2(2x − 6) − 2(2x − 3)
(2x − 6)
2
f′ (0) = − 1/6
f′ (a)(x − a) = −
x
6
f′ ′ (a)
2!
(x − a)
2
= −
x
2
18
f′ ′ ′ (a)
3!
(x − a)
3
= −
x
3
54
f (0) =
1
2
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Since the equation for the distribution of the planets would the solution of a differential equation,
the equation for the distribution being:
Then we integrate to obtain:
P
i
=
n
i
2 ⋅ 3
i
∫
n
i
2 ⋅ 3
i
di =
3
i
n
i
2log
(
n
3
)
+ C
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The AI Motif In Planetary Structure!
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It would seem the distribution of the planets have an AI motif. At this point I suggest
that it defies a single algebraic expression because it may be that it uses logic gate
arithmetic, which does not follow arithmetic as is done in mathematics, but rather in
creating logic circuits with the seven basic gates, from which all other gates can be
constructed. In other words, the structure of the solar system may be a circuit diagram. !
We have noted that the planets quantize as in binary, and that they take a pattern in
terms of semiconductor materials used to make binary digital logic gates. Here we
further note that while the planets distribute according to , so does the number of
rows in a truth table used to describe inputs and outputs of a logic gate if n is the
number of inputs. Let’s look at this a little.!
Digital logic is implemented by using a high input possibility and low input possibility to
achieve an ouput characteristic of the type of gate. Thus, for an AND gate, if one input
is high and one is low, the output is zero. If both are zero the output is zero. If both are
high the output is high. We say low is zero or false, and high is 1 or true. For an AND
gate, then, the truth table looks like:!
Where A and B are inputs and O is output. We say for an AND gate that O=AB. For an
OR Gate, a high input at A and no input at B results in and output at O. For a zero at A
and a 1 at B there is an output at O as well. For inputs at A and B there is an output at
O and for no inputs at A and B there is not output. The truth table looks like:!
And we say O=A+B!
The not gate only has one input which if high has a zero output and if is low has an
output. Its truth table is then:"
2
n
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!
And we say . These are three of the seven basic gates from which we can build
the others which are the NOR, NAND, XOR, and XNOR. For instance, the NOR gate
can be formed using the OR gate and NOT gate as seen in this truth table:!
Thus we have the following basic gates…"
O =
¯
A
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!
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Thus I would suggest the search for and expression for the planets not in terms of the algebra
we are all familiar with…!
A=B+C!
Z=XY!
ax+bx+cz=0!
But in terms of logic circuit mathematics using expressions that look like, for instance…!
!
I have already started with the inner terrestrial planets and we see that Mercury and Venus are
OR gates coupled with one another and Earth is an OR Gate coupled with Mars an AND gate.
Next two pages to see this…"
(A ⊕ B)(A ⊙ B) = C
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"
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"
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The Author!